Differential functional von Foerster equations with renewal
Natural iterative methods converge to the exact solution of a differential-functional von Foerster-type equation which describes a single population dependent on its past time and state densities as well as on its total size. On the lateral boundary we impose a renewal condition. Природнi iтератив...
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Інститут фізики конденсованих систем НАН України
2008
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| Цитувати: | Differential functional von Foerster equations with renewal / H. Leszczyński // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 361-370. — Бібліогр.: 15 назв. — англ. |
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Leszczyński, H. 2017-06-05T17:57:15Z 2017-06-05T17:57:15Z 2008 Differential functional von Foerster equations with renewal / H. Leszczyński // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 361-370. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 87.10.+e, 82.39.-k, 82.39.Rk DOI:10.5488/CMP.11.2.361 https://nasplib.isofts.kiev.ua/handle/123456789/119287 Natural iterative methods converge to the exact solution of a differential-functional von Foerster-type equation which describes a single population dependent on its past time and state densities as well as on its total size. On the lateral boundary we impose a renewal condition. Природнi iтеративнi методи збiгаються до точного розв’язку диференцiально-функцiонального рiвняння типу фон Фьорстера, що описує популяцiю, залежну вiд своїх минулих густин станiв i вiд загального розмiру. На бiчнiй границi ми накладаємо умову вiдновлення. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Differential functional von Foerster equations with renewal Диференцiальнi функцiональнi рiвняння фон Фьорстера з вiдновленням Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Differential functional von Foerster equations with renewal |
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Differential functional von Foerster equations with renewal Leszczyński, H. |
| title_short |
Differential functional von Foerster equations with renewal |
| title_full |
Differential functional von Foerster equations with renewal |
| title_fullStr |
Differential functional von Foerster equations with renewal |
| title_full_unstemmed |
Differential functional von Foerster equations with renewal |
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differential functional von foerster equations with renewal |
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Leszczyński, H. |
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Leszczyński, H. |
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2008 |
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English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
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Диференцiальнi функцiональнi рiвняння фон Фьорстера з вiдновленням |
| description |
Natural iterative methods converge to the exact solution of a differential-functional von Foerster-type equation
which describes a single population dependent on its past time and state densities as well as on its total size.
On the lateral boundary we impose a renewal condition.
Природнi iтеративнi методи збiгаються до точного розв’язку диференцiально-функцiонального рiвняння типу фон Фьорстера, що описує популяцiю, залежну вiд своїх минулих густин станiв i вiд загального розмiру. На бiчнiй границi ми накладаємо умову вiдновлення.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119287 |
| citation_txt |
Differential functional von Foerster equations with renewal / H. Leszczyński // Condensed Matter Physics. — 2008. — Т. 11, № 2(54). — С. 361-370. — Бібліогр.: 15 назв. — англ. |
| work_keys_str_mv |
AT leszczynskih differentialfunctionalvonfoersterequationswithrenewal AT leszczynskih diferencialʹnifunkcionalʹnirivnânnâfonfʹorsterazvidnovlennâm |
| first_indexed |
2025-11-26T17:39:48Z |
| last_indexed |
2025-11-26T17:39:48Z |
| _version_ |
1850765983932743680 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 2(54), pp. 361–370
Differential functional von Foerster equations with
renewal
H.Leszczyński
Univ. Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
Received January 31, 2008
Natural iterative methods converge to the exact solution of a differential-functional von Foerster-type equation
which describes a single population dependent on its past time and state densities as well as on its total size.
On the lateral boundary we impose a renewal condition.
Key words: iterative method, differential functional, Hale operator
PACS: 87.10.+e, 82.39.-k, 82.39.Rk
1. Introduction
The paper is dedicated to the memory of Professor A. Lasota. Von Foerster and Volterra-Lotka
equations arise in biology, medicine and chemistry, [1,7,11,12,14]. The independent variables xj
and the unknown function u stand for certain features and densities, respectively. It follows from
this natural interpretation that xj > 0 and u > 0. Von Foerster model is essentially nonlocal,
because it contains the total size of population
∫
u(t, x)dx.
Existence results for certain von Foerster type problems have been established by means of
the Banach contraction principle, the Schauder fixed point theorem or iterative methods, see [2]
and [3–5]. Nonlocal terms always cause huge problems. Satisfactory conditions for convergence of
iterative methods were provided in [9], where (for the sake of simplicity) some boundary data were
prescribed. This forced additional restrictions on the (tangential!) flow of bicharacteristics near the
lateral boundary.
In the present paper we generalize the L∞∩L1-convergence results of [9] to the case of renewal
boundary conditions with natural assumptions on the flow of bicharacteristics. An associate re-
sult to [9] on fast convergent quasi-linearization methods has been published in [8]. The renewal
condition of the form
u(t, 0) =
∫ ∞
0
k(t, x, y)u(t, x)dx
is interpreted as giving birth to young individuals at the age 0 by mature individuals (0 < x < +∞).
In A. Lasota’s investigations there were included some delay effects like
∫
u(t− r, x)dx. Such cases
are also available using the method of the present paper. Investigations of finite difference schemes
on large meshes with strongly nonlocal functionals were analyzed in [15]. Since the renewal case is
complex, we arrive at its reduction to the one with given data on the lateral boundary. In fact, the
unknown renewal part is expressed by a Green function. This approach can be extended to many
species models. There is also a difference between the renewal case and [9]: the renewal involves
such coupling of the unknown function that excludes monotone iterations.
1.1. Formulation of the differential problem.
Let τ = (τ1, . . . , τn) ∈ Rn
+, τ0 > 0, where R+ := [0,+∞). Define
B = [−τ0, 0] × [−τ, τ ], where [−τ, τ ] = [−τ1, τ1] × · · · × [−τn, τn]
c© H.Leszczyński 361
H.Leszczyński
and
E0 = [−τ0, 0] × Rn, E = [0, a] × Rn, (a > 0), ∂E = {(t, x) ∈ E : x1 · · · · · xn = 0} .
For each function w defined on [−τ0, a], we have the Hale functional wt (see [6]), which is the
function defined on [−τ0, 0] by
wt(s) = w(t+ s), (s ∈ [−τ0, 0]).
For each function u defined on E0 ∪E, we similarly write a Hale-type functional u(t,x), defined on
B by
u(t,x)(s, y) = u(t+ s, x+ y) for (s, y) ∈ B.
Let Ω0 = E × C ([−τ0, a],R+) and Ω = E × C (B,R+) × C ([−τ0, a],R+) . Take v : E0 → R+
and
cj : Ω0 → R, λ : Ω → R (j = 1, . . . , n).
Consider the differential-functional equation
∂u
∂t
+
n
∑
j=1
cj (t, x, z[u]t)
∂u
∂xj
= u(t, x)λ
(
t, x, u(t,x), z[u]t
)
, (1)
where
z[u](t) :=
∫
Rn
+
u(t, y)dy, t ∈ [−τ0, a], (2)
with the initial conditions
u(t, x) = v(t, x), (t, x) ∈ E0 , x = (x1, . . . , xn) ∈ Rn
+ , (3)
and with the renewal condition
u(t, x) =
∫
Rn
+
k(t, x, y)u(t, y)dy, (t, x) ∈ ∂E, y ∈ Rn
+ , (4)
where k : ∂E×Rn
+ → Rn
+. We are looking for Caratheodory’s solutions to (1)–(4), see [2] and [10].
The functional dependence includes a possible delayed and integral dependence of the Volterra
type. The Hale functional z[u]t takes into consideration the whole population within the time
interval [t− τ0, t], whereas the Hale-type functional u(t,x) shows the dependence on the density u
locally in a left neighbourhood of (t, x). For simplicity we assume in the paper that n = 1. Notice
that it is possible to extend the result to the case n > 1 with quite technical multiple integrals on
the lateral boundary.
2. Bicharacteristics
First, for a given function z ∈ C([−τ0, a],R+), consider the bicharacteristic equations for
problem (1), (3):
η′(s) = c (s, η(s), zs) , η(t) = x. (5)
Denote by η = η[z](·; t, x) = (η1[z](·; t, x), . . . , ηn[z](·; t, x)) the bicharacteristic curve passing
through (t, x) ∈ E, i.e., the solution to problem (5). We consider its maximal (left) existence
domain to be an interval [α(t, x), t], where α(t, x) = 0 or 0 < α(t, x) 6 t. This alternative splits E
into two parts E0[z] and E+[z]. Next, we consider the following equation
d
d s
u(s, η[z](s; t, x)) = u(s, η[z](s; t, x))λ(s, η[z](s; t, x), u(s,η[z](s;t,x)), zs) (6)
with the initial condition
u(0, η[z](0; t, x)) = v(0, η[z](0; t, x)) for (t, x) ∈ E0[z], (7)
362
Differential functional von Foerster equations with renewal
and (with the brief notation α = α(t, x))
u(α, η[z](α; t, x)) = ṽ(α, η[z](α; t, x)) for (t, x) ∈ E+[z]. (8)
In the latter equation (6) the existence of a suitable extension ṽ of v to the lateral boundary is
complicated. We discuss this topic later on.
For any given function z ∈ C([−τ0, a],R+), a solution of equation (6) along bicharacteristics
(5) is a solution of (1). The initial conditions (3) and (7) or (8) correspond to each other.
Assume that:
(V0) v ∈ CB (E0,R+) (non-negative, bounded and continuous function);
(V1) z[v] ∈ C ([−τ0, 0],R+), where
z[v](t) =
∫
Rn
v(t, x)dx;
(V2) the function v satisfies the Lipschitz condition
|v(t, x) − v(t, x̄)| 6 Lv‖x− x̄‖ on E0
with some constant Lv > 0;
(C0) cj : Ω0 → R+ are positive, continuous and
‖c(t, x, q) − c(t, x̄, q̄)‖ 6 Lc‖x− x̄‖ + L∗
c‖q − q̄‖.
A continuous function σ : [0, a] × R+ → R+ is said to be a Perron comparison function if
σ(t, 0) ≡ 0 and the differential problem y′ = σ(t, y), y(0) = 0 has the only zero solution. We call it
uniform if σ, multiplied by any positive constant, is also a Perron comparison function. We call it
monotone if σ is non-decreasing in the second variable.
(Λ0) λ : Ω → R is continuous in (t, x, w, q) and
|λ(t, x, w, q) − λ(t, x̄, w̄, q̄)| 6 Mλ σ(t, ‖x− x̄‖ + ‖w − w̄‖ + ‖q − q̄‖),
where σ : [0, a] × R+ → R+ is a monotone, uniform Perron comparison function;
(Λ1) there exists a function Lλ ∈ L1([0, a],R+), such that
λ(t, x, w, q) 6 Lλ(t)
for (t, x) ∈ E,w ∈ C(B,Rm
+ ), q ∈ C([−τ0, a],R
m
+ ).
Denote
W (t, x, w, q) = λ(t, x, w, q) + tr ∂xc(t, x, q)
for (t, x) ∈ E,w ∈ C(B,R+), q ∈ C([−τ0, a],R+), where tr ∂xc stands for the trace of the matrix
∂xc = [∂xk
cj ]j,k=1,...,n .
(W0) There exists MW ∈ R+, such that
|W (t, x, w, q) −W (t, x̄, w̄, q̄)| 6 MW σ(t, ‖x− x̄‖ + ‖w − w̄‖ + ‖q − q̄‖),
where σ : [0, a] × R+ → R+ is a monotone, uniform Perron comparison function.
(W1) There exists a function LW ∈ L1([0, a],R+), such that
W (t, x, w, q) 6 LW (t)
for (t, x) ∈ E,w ∈ C(B,R+), q ∈ C([−τ0, a],R+).
(K0) The kernel k : ∂E × Rn
+ → Rn
+ is bounded and continuous, and a · ‖k‖∞ · ‖c‖ < 1.
The latter condition states that the length of the interval [0, a] is sufficiently small. This as-
sumption is superfluous if k(t, x, y) = 0.
363
H.Leszczyński
Lemma 2.1 If the conditions (V0), (Λ1), (K0) are satisfied, then any solution u of equation (6)
has the estimate
0 6 u(t, x) 6 ‖v‖∞ exp
(
∫ t
0
Lλ(s)ds
)
on E0[z]
and
0 6 u(t, x) 6 ‖v‖∞ exp
(
∫ t
0
(Lλ(s) + LW (s))ds
)
·
‖k‖∞
1 − a · ‖k‖∞ · ‖c‖∞
on E+[z].
Proof. The first inequality is standard. The second one will be explained in the following
subsection. �
2.1. The fixed point equation.
Let
Z(t) = max
−τ06s60
‖v(s, ·)‖1 exp
(
∫ t
0
LW (s)ds
)
+t · ‖c‖∞ · ‖v‖∞ exp
(
∫ t
0
(Lλ(s) + LW (s))ds
)
·
‖k‖∞
1 − a · ‖k‖∞ · ‖c‖∞
(9)
where we put LW (s) = 0 for s ∈ [−τ0, 0], and
Z = {z ∈ C([−τ0, a],R+) : z(t) 6 Z(t)}. (10)
Consider the operator T : Z → Z given by the formula
T [z](t) =
∫
Rn
+
u[z](t, x)dx for t > 0, (11)
where u = u[z] ∈ C1(B,R+) is the solution of (6)–(7) with the initial condition u[z](t, x) = v(t, x)
on E0. The function u = u[z] has the following integral representation
u[z](t, x) = v(α, η(α)) exp
(
∫ t
α
λ
(
s, η(s), u(s,η(s)), zs
)
ds
)
, (12)
where η(s) = η[z](s; t, x) and α = α(t, x) (α depends on z). By Lemma 2.1, we write (11) in the
following way
T [z](t) =
∫
Rn
+
v(α, η(α)) exp
(
∫ t
α
λ
(
s, η(s), u(s,η(s)), zs
)
ds
)
dx (13)
for t > 0. The bicharacteristics admit the group property:
y = η[z](0; t, x) ⇐⇒ η[z](s; t, x) = η[z](s; 0, y),
that is: any bicharacteristic curve passing through the points (0, y) and (t, x) has the same value
at s ∈ [0, a].
If we change the variables y = η[z](0; t, x), then, by the Liouville theorem, the Jacobian
J = det
[
∂c
∂x
]
is given for α = α(t, x) by the formula
J(α; t, x) = exp
(
−
∫ t
α
tr ∂xc(s, ηi[z](s; 0, y), zs)ds
)
.
364
Differential functional von Foerster equations with renewal
Hence (13) can be written in the form
T [z](t) =
∫
Rn
+
v(0, y) exp
(
∫ t
0
W (s, η(s), u(s,η(s)), zs)ds
)
dy
+
∫
St
u(α, η(α; t, x)) exp
(
∫ t
α
λ(s, η(s), u(s,η(s)), zs)ds
)
dx, (14)
where η(s) = η[z](s; 0, y), α = α(t, x) and St is the set of x ∈ Rn
+ such that α(t, x) > 0.
Lemma 2.2 If the conditions (V0), (V1), (W1), (K0) are satisfied, then
0 6 T [z](t) 6 Z(t) < +∞ for t ∈ [0, a],
where Z is given by (9).
Proof. This assertion follows directly from (14) and Assumptions (V0), (V1) and (W1). �
The respective fixed point equation for bicharaceristics η = η[z] has the form
η(s; t, x) = x−
∫ t
s
c(ζ, η(ζ; t, x), zζ)dζ. (15)
Lemma 2.3 If Assumption (C0) is satisfied and z, z̄ ∈ Z, then
‖η[z](s; t, x) − η[z̄](s; t, x)‖ 6
∫ t
s
L∗
c‖zζ − z̄ζ‖e
Lc(ζ−s)dζ.
Lemma 2.4 Under the Assumptions (V0), (V1), (C0), (K0), for each z ∈ Z, there exists the
unique continuous function ṽ : ∂E → R+ which satisfies the renewal condition.
Proof. The renewal condition (4) for (t, x) ∈ ∂E can be rewritten as follows
u(t, x) =
∫
Rn
+\St
k(t, x, y)u(t, y) +
∫
St
k(t, x, y)u(t, y)dy.
The first term is a bounded operator of v, see (12). From (12), the second term is equal to
∫
St
k(t, x, y) v(α, η(α)) exp
(
∫ t
α
λ
(
s, η(s), u(s,η(s)), zs
)
ds
)
dy,
where η(s) = η[z](s; t, y) and α = α(t, y). By (K0) the second term is small (has the norm less
than 1). �
The above lemma explains the estimate of Lemma 2.1. The next statement is crucial in our
paper.
Lemma 2.5 Under all previous assumptions, any solution z of the fixed point equation for (14)
has the representation
z(t) =
∫
Rn
+
v(0, y)G(t, y; z)dy,
where the Green function G has the same estimate as the first kernel in (14), multiplied by some
constant.
Proof. The assertion follows from a Neumann series expansion for u aided by the renewal
condition. �
365
H.Leszczyński
3. The iterative method.
Define the iterative method by z(k+1) = T [z(k)] with an arbitrary function z(0) ∈ Z, where the
class Z is defined by (10). We prove its uniform convergence under natural assumptions on the
given functions. The algorithm splits into three stages:
1. finding bicharacteristics η(k) = η[z(k)], given by (15),
2. finding u(k) = u[z(k)] as a solution of (12),
3. calculating z(k+1) = T [z(k)] by means of (13) or (14). In this way there are given the integral
equations
η(k)(s; t, x) = x−
∫ t
s
c(ζ, η(k)(ζ; t, x), z
(k)
ζ )dζ ,
u(k)(t, x) = v(α, η(k)(α; t, x)) exp
(
∫ t
α
λ
(
Q(k)(s)
)
ds
)
,
z(k+1)(t) =
∫
Rn
v(0, y) exp
(
∫ t
0
W
(
R(k)(s)
)
ds
)
dy
+
∫
St
u(α, η(k)(α; t, x)) exp
(
∫ t
α
λ
(
Q(k)(s)
)
ds
)
where α = α(t, x) (depends on z(k)) and
Q(k)(s) =
(
s, η(k)(s; t, x), u
(k)
(s,η(k)(s;t,x))
, z(k)
s
)
, R(k)(s) =
(
s, η(k)(s; 0, y), u
(k)
(s,η(k)(s;0,y))
, z(k)
s
)
.
Theorem 3.1 If z(0) ∈ Z and Assumptions (V 0)–(V 2), (C0), (Λ0), (Λ1), (W0), (W1), (K0) are
satisfied and there are K ∈ R+, θ ∈ (0, 1] such that
σ(t, r) 6 K tθ−1p r1−1/p for p > 2, (16)
then the iterative method z(k+1) = T [z(k)] is well defined in Z and uniformly converges to the
unique fixed point z = T [z], locally, that is: on a sufficiently small [0, a].
Remark 3.1 The technical condition (16) is fulfilled in the Lipschitz case (σ(t, r) = Lr) as well
as the simplest nonlinear Perron comparison functions such as σ(t, r) = Lr ln (1 + 1/r). Its formu-
lation also includes weak singularities, e.g. σ(t, r) = t−1/2Lr and σ(t, r) = t−1/2Lr ln (1 + 1/r).
Proof. (of Theorem 3.1) Denote
∆z(k) = z(k+1) − z(k), ∆η(k) = η(k+1) − η(k), ∆u(k) = u(k+1) − u(k).
Then we have the estimates
‖∆η(k)(s; t, x)‖ 6
∫ t
s
L∗
c‖∆z
(k)
ζ ‖eLc(ζ−s)dζ ,
|∆u(k)(t, x)| 6 C‖∆η(k)(0; t, x)‖ exp
(
∫ t
0
Lλ(s)ds
)
+ C exp
(
∫ t
0
Lλ(s)ds
)
∫ t
0
Mλσ
(
s, P (k)(s; t, x)
)
ds,
|∆z(k+1)(t)| 6 C
∫ t
0
MWσ
(
s, P (k)(s; t, x)
)
ds,
where P (k)(s; t, x) = ‖∆η(k)(s; t, x)‖+ ‖∆u(k)‖s + ‖∆z(k)‖s. In the above estimate the constant C
is generic (depending on the data).
366
Differential functional von Foerster equations with renewal
Denote L̂λ =
∫ a
0
Lλ(s)ds and
Ψ(k)(s, t) = ψ̄(k)(s) + ψ(k)(s) +
∫ t
s
L∗
ce
Lcaψ(k)(ζ)dζ .
Consider the comparison equations
ψ̄(k)(t) = C
∫ t
0
L∗
ce
Lca+L̂λψ(k)(s)ds+ C eL̂λ
∫ t
0
Mλσ
(
s,Ψ(k)(s, t)
)
ds, (17)
ψ(k+1)(t) = C
∫ t
0
MWσ
(
s,Ψ(k)(s, t)
)
ds (18)
with ψ(0)(t) = Z(t) and
ψ̄(0)(t) = C exp
(
∫ t
0
LW (s)ds
)
+ C
∫ t
0
L∗
ce
Lca+L̂λZ(s)ds
+ CeL̂λ
∫ t
0
Mλσ
(
s, ψ̄(0)(s) + Z(s) +
∫ t
s
L∗
ce
LcaZ(ζ)dζ
)
ds. (19)
The remaining part of the proof is split into several auxiliary lemmas.
Lemma 3.1 Under the assumptions of Theorem 3.1 there is a0 ∈ (0, a] such that |∆u(k)(t, x)| 6
ψ̄(k)(t), |∆z(k)(t)| 6 ψ(k)(t),
‖∆η(k)(s; t, x)‖ 6
∫ t
s
L∗
ce
Lcaψ(k)(ζ)dζ
on [0, a0] × R+, and the sequences {ψ(k)}, {ψ̄(k)} are non-decreasing in k.
Lemma 3.2 Under the assumptions of Theorem 3.1 the estimate
∫ t
0
σ(s,Asl +Btl+1)ds 6 tl+θ−l/p pKθ−1
[
A
θ + l
+
B a
θ
]1−1/p
holds.
Lemma 3.3 Under the assumptions of Theorem 3.1 the sequences {ψ(k)} and {ψ̄(k)} tend uni-
formly to 0 as k → +∞.
Proof. With some constants M , M∗ and ca, dependent on the data, the equation
ψ̂(t) = M
∫ t
0
ψ̂(s)ds+M∗
∫ t
0
σ
(
s, ψ̂(s) + ca
∫ t
s
ψ̂(ζ)dζ
)
ds (20)
describes a comparison function ψ̂ with respect to ψ + ψ̄, where
ψ(t) = lim
k→∞
ψ(k)(t), ψ̄(t) = lim
k→∞
ψ̄(k)(t).
One can prove by induction on k = 0, 1, . . . that ψ̂(t) 6 Ĉkt
θ/2 and Ĉka
θ/2 → 0 as k → +∞,
provided that the interval [0, a] is sufficiently small. Take an arbitrary Ĉ0 which estimates ψ̂(t).
Applying Lemma 3.2 with p = 2 to (20), we get
ψ̂(t) 6 MtĈ0 +M∗tθ2Kθ−1
[
Ĉ0 (1 + caa)
θ
]1/2
6 tθ/2Ĉ1,
367
H.Leszczyński
where
Ĉ1 = Ma1−θ/2Ĉ0 + aθ/22Kθ−1
[
Ĉ0 (1 + caa)
θ
]1/2
.
Suppose that the desired estimate holds for some k > 1. Applying Lemma 3.2 with p = 2k to (20),
we get
ψ̂(t) 6 M
t1+kθ/2
1 + kθ/2
Ĉk +M∗t(k+1)θ/22Kθ−1
[
Ĉk
θ + kθ/2
+
caĈk
θ (1 + kθ/2)
]1−1/(2k)
,
hence ψ̂(t) 6 t(k+1)θ/2Ĉk+1, where
Ĉk+1 = MĈk
a1−θ/2
1 + kθ/2
+M∗ 2Kθ−1
[
Ĉk
θ + kθ/2
+
caĈk
θ (1 + kθ/2)
]1−1/(2k)
.
The constants Ĉk have an upper estimate of the form AQk, thus ψ̂(t) ≡ 0 in a neighbourhood of
0 (because ψ̂(t) 6 AQktkθ/2).
Lemma 3.4 Under the assumptions of Theorem 3.1 the sequences {z(k)}, {u(k)}, {η(k)} tend
uniformly to z, u[z], η[z] such that z = T [z].
Proof. We intend to find the following estimates
ψ(k)(t) 6 Ckt
lk , ψ̄(k)(t) 6 C̄kt
lk ,
where the series
∑
k Ckt
lk is convergent in a neighbourhood of 0. The assertion is seen if we replace
the comparison equations (17)–(18) by the inequalities
C̄kt
lk > C
∫ t
0
L∗
ce
Lca+L̂λCks
lkds+ CeL̂λ
∫ t
0
Mλσ
(
s, (Ck + C̄k)slk + L∗
ce
LcaCkt
lk+1/(lk + 1)
)
ds,
Ck+1t
lk+1 > C
∫ t
0
MWσ
(
s, (Ck + C̄k)slk + L∗
ce
LcaCkt
lk+1/(lk + 1)
)
ds
with suitable C0t
l0 > Z(a) and with some
C̄0 > aC L∗
ce
Lca+L̂λZ(a) + CeL̂λ Mλ
∫ a
0
σ
(
s, C̄0 + Z(a) + aL∗
ce
LcaZ(a)
)
ds.
If we put
l0 = 0, p0 = 2/θ, lk = kθ/2, pk = 4 k for k = 1, 2, . . .
and exploit Lemma 3.2, then Ck, C̄k can be defined as making the series
∑
k Ckt
lk convergent,
hence the series ψ(0) + ψ(2) + . . . uniformly converges, and z(k) has a limit, which is continuous.
Corollary 3.1 If Assumptions (V 0)–(V 2), (C0), (Λ0), (Λ1), (W0) and (W1) are satisfied, then
there exists a unique solution of problem (1)–(3), locally with respect to t.
3.1. The Lipschitz case.
Suppose that Assumptions (V 0)–(V 2), (C0), (Λ1) and (W1), (K0) formulated in Section 2, are
valid. We modify some assumptions on the functions λ and W as follows:
(Λ̃0) λ : Ω → R is continuous in (t, x, w, q) and there exists a function L ∈ L1([0, a],R+), such
that
|λ(t, x, w, q) − λ(t, x̄, w̄, q̄)| 6 L(t)(‖x− x̄‖ + ‖w − w̄‖ + ‖q − q̄‖);
368
Differential functional von Foerster equations with renewal
(W̃0) W : Ω → R and there exists a function L ∈ L1([0, a],R+), such that
|W (t, x, w, q) −W (t, x̄, w̄, q̄)| 6 L(t)(‖x− x̄‖ + ‖w − w̄‖ + ‖q − q̄‖).
Using the same notation as in the proof of Theorem 3.1, we have the estimates of the form
ψ(k)(t) 6
Qk+1
(
∫ t
0
∆(s)ds
)k
k !
, t ∈ [0, a],
where Q is a generic constant. Hence the sequences {ψ(k)} and {ψ̄(k)} tend uniformly to 0 as
k → +∞.
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369
H.Leszczyński
Диференцiальнi функцiональнi рiвняння фон Фьорстера з
вiдновленням
Г.Лещиньскi
Унiверситет Ґданська, Ґданськ, Польща
Отримано 31 сiчня 2008 р.
Природнi iтеративнi методи збiгаються до точного розв’язку диференцiально-функцiонального рiв-
няння типу фон Фьорстера, що описує популяцiю, залежну вiд своїх минулих густин станiв i вiд
загального розмiру. На бiчнiй границi ми накладаємо умову вiдновлення.
Ключовi слова: iтеративний метод, диференцiальний функцiонал, оператор Гейла
PACS: 87.10.+e, 82.39.-k, 82.39.Rk
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